Multiple rotors aircraft and control method

ABSTRACT

A multiple rotors aircraft and a control method thereof are provided. The control method comprises the following steps. First, current motion information of the multiple rotors aircraft is obtained. Then, at least one control gain is adjusted through a gain adjustment function according to the current motion information. The gain adjustment function conforms to a non-Lipschitzian characteristic, and at least one rotor of the multiple rotors aircraft is controlled according to the control gain. Therefore, the multiple rotors aircraft would be ensured that its flight attitude is toward a target position, and the expected result would be conformed rapidly.

BACKGROUND OF THE INVENTION

Field of the Invention

The invention relates to a control method of an aircraft, andparticularly relates to a multiple rotors aircraft based on anon-Lipschitzian characteristic and a control method thereof.

Description of Related Art

Along with quick development of technology, development cost of unmannedaerial vehicles (UAV) (which are also referred to as unmanned aircraftsystems (UAS), etc.) originally applied in military field is alsodecreased, and various electronic companies actively involve in the UAVmarket. Regarding applications of good and food delivery and sportsphotography, the electronic companies try to implement the aboveapplications through the UAVs recently. The UAV market is expected tobring a lot of job opportunities, and its economic output is limitless.

On the other hand, most of the UAVs on the market adopt a design ofmultiple rotors aircraft. The multiple rotors aircraft hascharacteristics of high mobility, simple design, high security, etc.,and is adapted to carry on operations in surveillance, reconnaissance,exploration, transportation, etc., in near ground environment.Generally, a change in a flight attitude of the aircraft may cause achange in a flight direction of the aircraft, so as to change adirection and a magnitude of a thrust force. However, most of thecurrent multiple rotors aircrafts have characteristics of highnonlinearity and coupling, which brings a great challenge in design of asystem controller, and parameter uncertainty causes a log of influenceson a dynamic behavior of the multiple rotors aircraft.

SUMMARY OF THE INVENTION

The invention is directed to a multiple rotors aircraft and a controlmethod thereof, in which control gains for driving rotors in themultiple rotor aircraft are adjusted based on a convergence property ofdifferential equations, so as to achieve fast, stable and robusteffects.

The invention provides a control method, which is adapted to a multiplerotors aircraft, and includes following steps. Current motioninformation of the multiple rotors aircraft is obtained. Then, at leastone control gain is adjusted through a gain adjustment functionaccording to the current motion information, so as to control at leastone rotor of the multiple rotors aircraft according to the control gain.The gain adjustment function conforms to a non-Lipschitziancharacteristic.

In an embodiment of the invention, after the step of adjusting the atleast one control gain through the gain adjustment function according tothe current motion information, the method further includes adjusting atarget attitude parameter through an attitude estimation functionaccording to the current motion information and the control gain, wherethe attitude estimation function is

${\theta_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}\left( {{{- K_{x}}S_{x}} + {\overset{¨}{x}}_{d} - {k_{x}{\overset{.}{e}}_{x}} - {{h_{x}}S_{x}}} \right)}},{\varphi_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}{\left( {{{- K_{y}}S_{y}} + {\overset{¨}{y}}_{d} - {k_{y}{\overset{.}{e}}_{y}} - {{h_{y}}S_{y}}} \right).}}}$

In an embodiment of the invention, before the step of adjusting the atleast one control gain through the gain adjustment function according tothe current motion information, the method further includes defining anerror function, where the error function is

${{\overset{.}{S}}_{\xi} = {{{\overset{\sim}{U}}_{1}{{bm}^{- 1}\begin{bmatrix}{\theta_{d} + {\overset{\sim}{H}}_{x}} \\{\varphi_{d} + {\overset{\sim}{H}}_{y}} \\{\cos \; {\theta cos}\; \varphi}\end{bmatrix}}} + \begin{bmatrix}{{- {\overset{¨}{x}}_{d}} + {k_{x}{\overset{.}{e}}_{x}}} \\{{- {\overset{¨}{y}}_{d}} + {k_{y}{\overset{.}{e}}_{y}}} \\{{- {\overset{¨}{z}}_{d}} + {k_{z}{\overset{.}{e}}_{z}}}\end{bmatrix} + \begin{bmatrix}0 \\0 \\{- g}\end{bmatrix}}},$

b is a length constant, and the error function conforms to thenon-Lipschitzian characteristic.

In an embodiment of the invention, the current motion informationfurther includes a current position, the step of obtaining the currentmotion information of the multiple rotors aircraft includes obtainingthe current position, a current pitch angle, a current yaw angle and acurrent roll angle through at least one inertial navigation element.

According to another aspect, the invention provides a multiple rotorsaircraft, which includes at least one rotor, at least one inertialnavigation element and a processing unit. The rotors are respectivelycontrolled by at least one control gain. The inertial navigation elementis configured to obtain current motion information of the multiplerotors aircraft. The processing unit is coupled to the rotors and theinertial navigation element, and obtains the current motion informationof the multiple rotors aircraft through the inertial navigation element,and adjusts the at least one control gain through a gain adjustmentfunction according to the current motion information. The gainadjustment function conforms to a non-Lipschitzian characteristic.

In an embodiment of the invention, the non-Lipschitzian characteristicrepresents that a convergence value of a function is only zero, and theconvergence value is not varied after being converged to zero.

In an embodiment of the invention, the current motion informationincludes a current pitch angle, a current yaw angle and a current rollangle, and the gain) {dot over (h)}_(x)=−h_(x)^((1/2 n+1))[Ũ₁(−θ_(d)+cos ψ sin θ cos φ+sin ψ sin φ)S_(x)] adjustmentfunction is {dot over (h)}_(y)=−h_(y) ^(1/2n−1))[Ũ₁(−φ_(d)+sin ψ sin θcos φ−cos ψ sin φ)S_(y)], where

${\overset{.}{h}}_{i} = \left\{ \begin{matrix}{{{- h_{i}^{(\frac{1}{{2n} + 1})}}\Theta_{i}},{h_{i} \neq 0},{i = x},y} \\{\sigma,{h_{i} = 0}}\end{matrix} \right.$

h_(x), b_(y) are control gains,

${{\overset{\sim}{U}}_{1} = {\frac{1}{\cos \; {\theta cos}\; \varphi}\left( {g + {\overset{¨}{z}}_{d} - {k_{z}{\overset{.}{e}}_{z}} - {K_{z}S_{z}}} \right){\forall{{\theta } < \frac{\pi}{2}}}}},{{\varphi } < \frac{\pi}{2}},,\theta_{d}$

is a target pitch angle in a target attitude parameter, φ_(d) is atarget yaw angle in the target attitude parameter, ψ is the current rollangle, θ is the current pitch angel, φ is the current yaw angle,S_(x)=ė_(x) +k_(x)e_(x), e_(x)=x−x_(d) , S_(y)=ė_(y)+k_(y)e_(y),e_(y)=x−x_(d) , S_(z)=ė_(z)+k_(z)e_(z), e_(z) =z−z_(d),(x,y,z) is acurrent position, (x_(d),y_(d),z_(d)) is a target position, k_(x),k_(y), and k_(z) are control coefficients, K_(z) is an aerodynamicdamping coefficient, n is hierarchy, g is the acceleration of gravity,Θ_(i)=Ũ₁{tilde over (H)}_(i)S_(i), {tilde over (H)}_(x)=−θ_(d)+cos ψ sinθ cos φ+sin ψ sin φ, {tilde over (H)}_(y)=−φ_(d)+sin ψ sin θ cos φ−cos ψsin φ, and σ is a micro constant.

In an embodiment of the invention, the processing unit adjusts thetarget attitude parameters through an attitude estimation functionaccording to the current motion information and the control gain, wherethe attitude estimation function is

${\theta_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}\left( {{{- K_{x}}S_{x}} + {\overset{¨}{x}}_{d} - {k_{x}{\overset{.}{e}}_{x}} - {{h_{x}}S_{x}}} \right)}},{\varphi_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}{\left( {{{- K_{y}}S_{y}} + {\overset{¨}{y}}_{d} - {k_{y}{\overset{.}{e}}_{y}} - {{h_{y}}S_{y}}} \right).}}}$

In an embodiment of the invention, the processing unit defines an errorfunction, where the error function is

${{\overset{.}{S}}_{\xi} = {{{\overset{\sim}{U}}_{1}{{bm}^{- 1}\begin{bmatrix}{\theta_{d} + {\overset{\sim}{H}}_{x}} \\{\varphi_{d} + {\overset{\sim}{H}}_{y}} \\{\cos \; {\theta cos}\; \varphi}\end{bmatrix}}} + \begin{bmatrix}{{- {\overset{¨}{x}}_{d}} + {k_{x}{\overset{.}{e}}_{x}}} \\{{- {\overset{¨}{y}}_{d}} + {k_{y}{\overset{.}{e}}_{y}}} \\{{- {\overset{¨}{z}}_{d}} + {k_{z}{\overset{.}{e}}_{z}}}\end{bmatrix} + \begin{bmatrix}0 \\0 \\{- g}\end{bmatrix}}},$

b is a length constant, and the error function conforms to thenon-Lipschitzian characteristic.

According to the above descriptions, in the multiple rotors aircraft andthe control method thereof provided by the invention, the errorfunction, the gain adjustment functions and the attitude estimationfunctions are determined according to the non-Lipschitziancharacteristic, so as to adjust the control gains used for driving therotors and the target attitude parameters. In this way, the embodimentof the invention may accurately control the multiple rotors aircraft toreach a target position, so as to provide a robust adaptive controlsystem.

In order to make the aforementioned and other features and advantages ofthe invention comprehensible, several exemplary embodiments accompaniedwith figures are described in detail below.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding of the invention, and are incorporated in and constitute apart of this specification. The drawings illustrate embodiments of theinvention and, together with the description, serve to explain theprinciples of the invention.

FIG. 1 is a component block diagram of a multiple rotors aircraftaccording to an embodiment of the invention.

FIG. 2 is a flowchart illustrating a control method according to anembodiment of the invention.

FIG. 3A is a schematic diagram of a four-rotor aircraft according to anembodiment of the invention.

FIG. 3B is a coordinate schematic diagram.

FIG. 4A-FIG. 4C are respectively schematic diagrams of a current pitchangle, a current yaw angle and a current roll angle.

FIG. 5 is an example of a control flow according to an embodiment of theinvention.

DESCRIPTION OF EMBODIMENTS

In conditions of stability analysis of conventional differentialequations, it is known that exponential solutions are convergent. Amultiple rotors aircraft and a control method thereof provided by theembodiments of the invention are adapted to quickly estimate an actualstate (for example, a flight attitude) through specific differentialequations, so as to achieve a quick convergence effect. Moreover,through stability analysis, stability of the system is proved. Aplurality of embodiment complying with the spirit of the invention isprovided below, and users may suitably adjust the embodiments accordingto an actual requirement without being limited to the followingdescribed content.

FIG. 1 is a component block diagram of a multiple rotors aircraftaccording to an embodiment of the invention. Referring to FIG. 1, themultiple rotors aircraft 100 includes at least one set of rotors 110₁-110 _(N), at least one inertial navigation element 130 and aprocessing unit 150, where N is a positive integer (for example, 4, 6,8, etc.). The multiple rotors aircraft 100 can be multirotor aircraft ofany driving types, which is not limited by the invention.

Each of the rotors 110 ₁-110 _(N) may at least include (but not limitedto) components of blades, a blade hub, propellers, a torque arm, a rotorshaft, etc. The rotors 110 ₁-110 _(N) are respectively driven by acontrol gain, and the multiple rotors aircraft 100 may accordinglyimplement operations of hovering, maintaining attitude, levelling off,etc. Based on a different number of the rotors 110 ₁-110 _(N), themultiple rotors aircraft 100 may have a different term (for example, aquacioter, a hexacopter, an oetocopter, etc.), and the number of therotors 110 ₁-110 _(N) is not limited by the invention.

The inertial navigation element 130 (or an inertial measurement unit(IMU)) includes an accelerometer, a gyro sensor, an E-compass, aG-sensor, a geomagnetic sensor, etc., and is configured to obtaincurrent motion information (for example, a current position, a currentpitch angle, a current yaw angle, a current roll angle, etc.) of themultiple rotors aircraft 100. It should be noted that according to adifferent design requirement, the multiple rotors aircraft 100 may alsoinclude a satellite navigation system such as a global positioningsystem (GPS), an assisted global positioning system (AGPS), a Galileopositioning system or a global navigation satellite system (GLONASS),etc., to assist obtaining position of geodesic coordinates (orgeographical position).

The processing unit 150 is coupled to the rotors 110 ₁-110 _(N) and theinertial navigation element 130, and is a central processing unit (CPU),or other programmable general purpose or special purpose microprocessor,a digital signal processor (DSP), a programmable controller, anapplication specific integrated circuit (ASIC) or other similar devicesor a combination of the above devices. In the present embodiment, theprocessing unit 150 is configured to execute all of operations of themultiple rotors aircraft 100. For example, the processing unit 150 has aproportional-integral-differential (PID) controller, which controlsrotation speeds of the rotors 110 ₁-110 _(N) based on the control gain,and adjusts the control gains and a flight attitude, etc., according toa control method of the invention, which is described in detail later.

It should be noted that according to a different design requirement, themultiple rotors aircraft 100 may further include a motor (for example, abrushless direct current (BLDC) motor, an electric motor, a frame, abracket, a power module, etc., and appearances, configuration positionsand sizes of the above components are not limited by the invention.

In order to facilitate understanding of an operation flow of theinvention, a plurality of embodiments is provided below to describe acontrol method of the multiple rotors aircraft 100 of the invention indetail. FIG. 2 is a flowchart illustrating a control method according toan embodiment of the invention. Referring to FIG. 2, the control methodof the present embodiment is adapted to the multiple rotors aircraft 100of FIG. 1. In the following description, the control method of thepresent embodiment is described with reference of various components andmodules of the multiple rotors aircraft 100. The flow of the controlmethod can be adjusted according to an actual requirement, and is notlimited to the following implementation.

In step S210, the processing unit 150 obtains current motion informationof the multiple rotors aircraft 100 through the inertial navigationelement 130. In the present embodiment, the current motion informationof the multiple rotors aircraft 100 includes a current pitch angle φ, acurrent yaw angle θ, a current roll angle ψ (or referred to as an Eulerangle coefficient in a rotation sequence E_(y)-E_(z)-E_(x)) and acurrent position. To be specific, in order to further understand theflight attitude of the multiple rotors aircraft 100, a dynamic model ofa four-rotor aircraft is taken as an example for preliminary derivation.FIG. 3A is a schematic diagram of a four-rotor aircraft according to anembodiment of the invention. The four-rotor aircraft 300 has componentsthat are the same or similar to that of the multiple rotors aircraft 100of FIG. 1, and detailed descriptions of the components are not repeated.Referring to a coordinate schematic diagram of FIG. 3B, the coordinatedefinition of the four-rotor aircraft 300 can be represented as follows:q=(x, y, z, φ, θ, ψ) ∈ R⁶, where (x,y,z) represents the current positionof a center of the four-rotor aircraft 300 relative to the geodesiccoordinates (or geographical coordinates), and (φ, θ, ψ) are threerotation angles (i.e. the current pitch angle φ, the current yaw angle θand the current roll angle ψ) used for describing the flight attitude ofthe four-rotor aircraft 300.

FIG. 4A-FIG. 4C are respectively schematic diagrams of the current pitchangle φ, the current yaw angle θ and the current roll angle ψ. Referringto FIG. 3B and FIG. 4A, taking the geodesic coordinates(E_(x)-E_(y)-E_(z)) as a basis, an included angle between an E_(x) axisand a connection line of the rotors 310 ₁ and 310 ₃ towards an E_(z)axis direction is defined as the current pitch angle φ. Referring toFIG. 3B and FIG. 4B, taking the geodesic coordinates (E_(x)-E_(y)-E_(z))as the basis, an included angle between an E_(y) axis and a connectionline of the rotors 310 ₂ and 310 ₄ towards the E_(z) axis direction isdefined as the current yaw angle θ. Referring to FIG. 3B and FIG. 4C, anincluded angle between the rotors 310 ₁ and 310 ₄ is defined as thecurrent roll angle ψ.

The processing unit 150, for example, obtains sensing information (forexample, an azimuth angle, an acceleration, a speed, a displacement,etc.) through the inertial navigation element 130 such as a gyro sensor,an accelerometer, an E-compass, etc., and calculates the currentposition and the current pitch angle φ, the current yaw angle θ and thecurrent roll angle ψ according to the sensing information.

Moreover, a basic principle of navigation is to ensure correcttransformation of two coordinate systems (i.e. on-vehicle coordinatesand the geodesic coordinates). Then, referring to FIG. 3B, B={B₁, B₂,B₃} is defined as the on-vehicle coordinates, and the on-vehiclecoordinates can be transformed into the geodesic coordinates through afollowing transformation matrix R_(E):

${R_{E} = \begin{bmatrix}{\cos \; {\theta cos\psi}} & {\sin \; {\psi sin}\; \theta} & {{- \sin}\; \theta} \\{\cos \; {\psi sin}\; {\theta sin}\; \varphi} & {{\sin \; {\psi sin}\; \theta \; \sin \; \varphi} + {\cos \; {\psi cos}\; \varphi}} & {\cos \; {\theta cos}\; \varphi} \\{{\cos \; {\psi sin}\; \theta \; \cos \; \varphi} + {\sin \; {\psi sin}\; \varphi}} & {{\sin \; {\psi sin}\; \theta \; \cos \; \varphi} - {\cos \; {\psi sin}\; \varphi}} & {\cos \; {\theta cos}\; \varphi}\end{bmatrix}},$

and a dynamic model can be defined as

${\overset{¨}{x} = {{\frac{1}{m}\left( {{\cos \; {\psi sin}\; {\theta cos}\; \varphi} + {\sin \; {\psi sin}\; \varphi}} \right)U_{1}} + \frac{A_{x}}{m}}},{\overset{¨}{y} = {{\frac{1}{m}\left( {{\sin \; {\psi sin}\; {\theta cos}\; \varphi} + {\cos \; {\psi sin}\; \varphi}} \right)U_{1}} + \frac{A_{y}}{m}}},{\overset{¨}{z} = {{- g} + {\frac{1}{m}\left( {\cos \; {\theta cos\varphi}} \right)} + \frac{A_{y}}{m}}},$

where m is a mass of the four-rotor aircraft 300, g is an accelerationof gravity (9.8 km/s), and [A_(x) A_(y) A_(z)] is an aerodynamic forcesvector.

It should be noted that in an actual practice, the processing unit 150may directly obtain a geodesic position (or a geographical position)through the satellite positioning system. In order to ensure the correcttransformation of the two coordinate systems, the processing unit 150requires to perform calibration at all the time. For example, theprocessing unit 150 corrects the current roll angle ψ based on theE-compass and the geomagnetic sensor, corrects the current pitch angle φand the current yaw angle θ based on the accelerometer and the G-sensor.After the correct transformation of the two coordinate systems isensured, the processing unit 150, for example, performs an integraloperation based on the gyro sensor to obtain attitude information basedon the on-vehicle coordinate system (i.e. the current pitch angle φ, thecurrent yaw angle θ and the current roll angle ψ).

In step S220, the processing unit 150 adjusts a control gain through again adjustment function according to the current motion information, soas to control the rotors 110 ₁-110 _(N) of the multiple rotors aircraft100 through the control gain. The gain adjustment function conforms to anon-Lipschitzian characteristic. The non-Lipschitzian characteristicrepresents that a convergence value of the function is only zero, andthe convergence value is varied after being converged to zero. To bespecific, derivation of the gain adjustment function is described below.First, a translational kinetic energy T_(trans) and a rotation kineticenergy T_(rot) can be respectively represented by following equations(1) and (2):

$\begin{matrix}{T_{trans} = {\frac{m}{2}{\overset{.}{\xi}}^{T}\overset{.}{\xi}}} & (1) \\{T_{rot} = {\frac{m}{2}{\overset{.}{\eta}}^{T}J\overset{.}{\eta}}} & (2)\end{matrix}$

Where, ζ=(x, y, z) ∈ R³ , η=(φ, θ, ψ) ∈ S³ , J is a rotation inertiamatrix. A potential energy U of the four-rotor aircraft 300 in FIG. 3Acan be represented by a following equation (3):

U=mgz   (3)

A Lagrangian function can be represented in a following equation (4):

$\begin{matrix}{{{\frac{}{t}\frac{\partial L}{\partial\overset{.}{q}}} - \frac{\partial L}{\partial q}} = F} & (4)\end{matrix}$

Where, q=[φ θ ψ]′. By applying the above equations (1)-(3) to L(q,{dotover (q)}), a following equation (5) is obtained:

$\begin{matrix}{{{L\left( {q,\overset{.}{q}} \right)} = {{T_{trans} + T_{rot} - U} = {{\frac{m}{2}{\overset{.}{\xi}}^{T}\overset{.}{\xi}} + {\frac{m}{2}{\overset{.}{\eta}}^{T}J\overset{.}{\eta}} - {mgz}}}},} & (5) \\{and} & \; \\{F = \begin{bmatrix}F_{\zeta} \\\tau\end{bmatrix}} & (6)\end{matrix}$

Where,

$\begin{matrix}{F_{\zeta} = {\quad{{\begin{bmatrix}{\cos \; {\theta cos\psi}} & {\sin \; {\psi sin}\; \theta} & {{- \sin}\; \theta} \\{\cos \; {\psi sin}\; {\theta sin}\; \varphi} & \begin{matrix}{{\sin \; {\psi sin}\; \theta \; \sin \; \varphi} +} \\{\cos \; {\psi cos}\; \varphi}\end{matrix} & {\cos \; {\theta cos}\; \varphi} \\\begin{matrix}{{\cos \; {\psi sin}\; \theta \; \cos \; \varphi} +} \\{\sin \; {\psi sin}\; \varphi}\end{matrix} & \begin{matrix}{{\sin \; {\psi sin}\; \theta \; \cos \; \varphi} -} \\{\cos \; {\psi sin}\; \varphi}\end{matrix} & {\cos \; {\theta cos}\; \varphi}\end{bmatrix}\begin{bmatrix}0 \\0 \\{f_{1} + f_{2} + f_{3} + f_{4}}\end{bmatrix}},}}} & (7) \\{\mspace{79mu} {and}} & \; \\{\mspace{79mu} {\tau = {\begin{bmatrix}\tau_{\varphi} \\\tau_{\theta} \\\tau_{\psi}\end{bmatrix} = \begin{bmatrix}{\left( {f_{1} - f_{3}} \right)l} \\{\left( {f_{4} - f_{2}} \right)l} \\\left( {{- f_{1}} - f_{3} + f_{2} + f_{4}} \right)\end{bmatrix}}}} & (8)\end{matrix}$

Where, f₁, f₂, f₃, f₄ are respectively thrust forces (corresponding to aB₃ axis direction of the on-vehicle coordinate system, or a directionperpendicular to the blades of the rotors 310 ₁, 310 ₂, 310 ₃, 310 ₄) ofthe rotors 310 ₁, 310 ₂, 310 ₃, 310 ₄, l is a length from a motor shaftto a mass center.

By organizing the above equations, following equations (9) and (10) areobtained:

$\begin{matrix}{{m\; \overset{¨}{\xi}} = {{\left( {f_{1} + f_{2} + f_{3} + f_{4}} \right)\begin{bmatrix}{{- \sin}\; \theta} \\{\cos \; {\theta sin}\; \varphi} \\{\cos \; {\theta cos}\; \varphi}\end{bmatrix}} + \begin{bmatrix}0 \\0 \\{mg}\end{bmatrix}}} & (9) \\{{J\; \overset{¨}{\eta}} = {{{- {C\left( {\eta,\overset{.}{\eta}} \right)}}\overset{.}{\eta}} + \tau}} & (10)\end{matrix}$

Where, C(η,{dot over (η)})) is a Coriolis term.

In order to overcome a rotation moment produced when the four sets ofrotors 310 ₁, 310 ₂, 310 ₃, 310 ₄ simultaneously rotate, the four-rotoraircraft 300 may adopt a symmetric rotation direction design (forexample, rotation directions M₁, M₂, M₃, M₄ of the rotors 310 ₁, 310 ₂,310 ₃, 310 ₄ in FIG. 3A). Based on a special design, an influence of therotation moment on the four-rotor aircraft 300 produced when the rotors310 ₁, 310 ₂, 310 ₃, 310 ₄ simultaneously rotate can be decreased. Ifthe four-rotor aircraft 300 is required to have a variation in thecurrent pitch angle φ, the thrust forces are designed to satisfy f₁>f₃or f₁<f₃, and meanwhile a position of the four-rotor aircraft 300 on theE_(x) axis of the geodesic coordinates is also changed (as shown in FIG.4A). If the four-rotor aircraft 300 is required to have a variation inthe current yaw angle θ, the thrust forces are designed to satisfy f₄>f₂or f₄<f₂, and meanwhile a position of the four-rotor aircraft 300 on theE_(y) axis of the geodesic coordinates is also changed (as shown in FIG.4B). FIG. 4C presents a symmetric thrust force to produce a variation inthe current roll angle ψ, and meanwhile a height of the aircraft ischanged. Therefore, by changing an attitude of the aircraft, a flightdirection of the aircraft is changed, and meanwhile the change of theattitude of the aircraft causes a change in a direction and a magnitudeof the thrust force.

Therefore, deduced according to a robust adaptive control incollaboration with the non-Lipschitzian characteristic (i.e. the finitetime is converged to zero), it is known that in the technique of themultiple rotors aircraft, a position control is a rather important link.A main purpose of the position control is to control the currentposition of the multiple rotors aircraft 100 to move to a targetposition (x_(d), y_(d), z_(d)) within a certain time. Since theprocessing unit 150 is required to calculate target attitude parameters(a target pitch angle θ_(d), a target yaw angle φ_(d), a target rollangle ψ_(d)) required by the multiple rotors aircraft 100 according tothe required target position during the calculation process, so as todrive the multiple rotors aircraft 100 to move to the target position,(to facilitate description, the four-rotor aircraft 300 of FIG. 3A istaken as an example for description) each axis acceleration can bedefined as a following equation (11):

$\begin{matrix}{\begin{bmatrix}\overset{¨}{x} \\\overset{¨}{y} \\\overset{¨}{z}\end{bmatrix} = {{m^{- 1}{b\begin{bmatrix}{\left( {\Omega_{1}^{2} + \Omega_{2}^{2} + \Omega_{3}^{2} + \Omega_{4}^{2}} \right)\theta_{d}} \\{\left( {\Omega_{1}^{2} + \Omega_{2}^{2} + \Omega_{3}^{2} + \Omega_{4}^{2}} \right)\varphi_{d}} \\{\left( {\Omega_{1}^{2} + \Omega_{2}^{2} + \Omega_{3}^{2} + \Omega_{4}^{2}} \right)\cos \; {\theta cos}\; \varphi}\end{bmatrix}}} + {\quad\begin{bmatrix}{m^{- 1}{b\left( {\Omega_{1}^{2} + \Omega_{2}^{2} + \Omega_{3}^{2} + \Omega_{4}^{2}} \right)}{\overset{\sim}{H}}_{x}} \\{m^{- 1}{b\left( {\Omega_{1}^{2} + \Omega_{2}^{2} + \Omega_{3}^{2} + \Omega_{4}^{2}} \right)}{\overset{\sim}{H}}_{y}} \\{- g}\end{bmatrix}}}} & (11)\end{matrix}$

Where, Ω₁, Ω₂, Ω₃, Ω₄ are respectively rotation speeds of the four setsof rotors in the four-rotor aircraft 300, b is a frame related lengthcoefficient, and {tilde over (H)}_(x), {tilde over (H)}_(y) arerespectively defined as following equations (12), (13):

{tilde over (H)} _(x)=−θ_(d)+cos ψ sin θ cos φ+sin ψ sin φ  (12)

{tilde over (H)} _(y)=−φ_(d)+sin ψ sin θ cosφ−cos ψ sin φ  (13)

In order to ensure the multiple rotors aircraft 100 to fly to the targetposition, a position tracking trajectory error function is defined as afollowing equation (14):

S _(ζ) =ė _(ζ) +k _(ζ) e _(ζ)  (14)

Where, S₇₀=└S_(x) S_(y) S_(z)┘=└ė_(x)+k_(x)e_(x) ė_(y)+k_(y)e_(y)ė_(z)+k_(z)e_(z)┘ and e_(ζ)=[e_(x) e_(y) e_(z)]^(T)=[x−x_(d) y−y_(d)z−z_(d)]^(T), and

${k_{\xi} = \begin{bmatrix}k_{x} & 0 & 0 \\0 & k_{y} & 0 \\0 & 0 & k_{z}\end{bmatrix}},$

k_(x), k_(y), and k_(z) are control coefficients (real numbers).

According to the position tracking trajectory error function (14), it isknown that when S_(ζ)=0, it represents that the error dynamicallyreaches a sliding surface, and such result is similar to a variablestructure system well known by those skilled in the art, and each timewhen the error reaches the sliding surface to make S_(ζ)=0, the trackingtrajectory error is ensured to be zero, so as to achieve an effect ofposition control.

Then, the target position ζ_(d)(t) is set as a bounded function, and adifferential of the target potion (i.e. the target speed) ζ_(d)^((i))(t) ∈ R, i=1,2,3 is bounded and conforms to ζ_(d) ^((i))(t) ∈L_(∞).

It should be noted that a main design purpose of the position control isto make the current position of the aircraft to reach the targetposition, i.e.

${\lim\limits_{t->\infty}{e_{\xi}}} = 0.$

Therefore, a primary condition is to ensure the processing unit 150 toobtain the current position of the multiple rotors aircraft 100 (i.e.the step S210). Then, the processing unit 150 defines an error function(15) as follows:

$\begin{matrix}{{{\overset{.}{S}}_{\xi} = {{{\overset{\sim}{U}}_{1}{{bm}^{- 1}\begin{bmatrix}{\theta_{d} + {\overset{\sim}{H}}_{x}} \\{\varphi_{d} + {\overset{\sim}{H}}_{y}} \\{\cos \; \theta \; \cos \; \varphi}\end{bmatrix}}} + \begin{bmatrix}{{- {\overset{¨}{x}}_{d}} + {k_{x}{\overset{.}{e}}_{x}}} \\{{- {\overset{¨}{y}}_{d}} + {k_{y}{\overset{.}{e}}_{y}}} \\{{- {\overset{¨}{z}}_{d}} + {k_{z}{\overset{.}{e}}_{z}}}\end{bmatrix} + \begin{bmatrix}0 \\0 \\{- g}\end{bmatrix}}},} & (15)\end{matrix}$

and defines Ũ₁ as the main thrust force of the multiple rotors aircraft100. For example, the main thrust force of the four-rotor aircraft 300of FIG. 3A is Ũ₁=Ω₁ ²+Ω₂ ²+Ω₃ ²+Ω₄ ². The error function (15) conformsto the non-Lipschitzian characteristic.

It should be noted that rules of a non-Lipschitzian autonomous systeminclude following theorems. Considering that the autonomous system {dotover (x)}=f(x), where f:D→R^(n) is non-Lipschitzian continuous on anorigin open neighborhood D ⊂ R^(n) of an origin of R^(n). The origin of{dot over (x)}=f(x) is a finite time convergence if there exists anorigin N ⊂ D of the origin and a function T_(x):N\{0}→(0, ∞) calls asetting time function such that every solution a trajectory x(t, x₀) of{dot over (x)}=f(x) starting from an initial point x₀ ∈ 8 0, T_(x)(x₀)),and

${{\lim\limits_{t->{T_{x}{(x_{0})}}}{x\left( {t,x_{0}} \right)}} = 0},.$

In other words, a condition of a differential equation beingdifferentiable is that it is continuous and smooth. If the uniquesolution to the differential equation is 0 and continuous but notsmooth, it is undifferentiable. However, if the solution to thedifferential equation does not vary after reaching 0, it indicates thatthe differential equation converges to 0.

Then, based on height control, the main thrust force can be modified toa following equation (16):

$\begin{matrix}{{{\overset{\sim}{U}}_{1} = {\frac{1}{\cos \; \theta \; \cos \; \varphi}\left( {g + {\overset{¨}{z}}_{d} - {k_{z}{\overset{.}{e}}_{z}} - {K_{z}S_{z}}} \right){\forall{{\theta } < {\pi/2}}}}},{{\varphi } < {\pi/2}},} & (16)\end{matrix}$

Where, K_(z) is an aerodynamic damping coefficient (a real number).

Regarding the control gains for the E_(x) axis and the E_(y) axis, thegain adjustment functions (17), (18), (19) can be deduced based on thenon-Lipschitzian characteristic, a Kalman filter and a sliding modeobserver:

$\begin{matrix}{{\overset{.}{h}}_{x} = {- {h_{x}^{({{{1/2}n} + 1})}\left\lbrack {{{\overset{\sim}{U}}_{1}\left( {{- \theta_{d}} + {\cos \; {\psi sin}\; {\theta cos}\; \varphi} + {\sin \; \psi \; \sin \; \varphi}} \right)}S_{x}} \right\rbrack}}} & (17) \\{{\overset{.}{h}}_{y} = {- {h_{y}^{({{{1/2}n} + 1})}\left\lbrack {{{\overset{\sim}{U}}_{1}\left( {{- \varphi_{d}} + {\sin \; {\psi sin}\; \theta \; \cos \; \varphi} - {\cos \; \psi \; \sin \; \varphi}} \right)}S_{y}} \right\rbrack}}} & (18) \\{{\overset{.}{h}}_{i} = \left\{ \begin{matrix}{{{- h_{i}^{({{{1/2}n} + 1})}}\Theta_{i}},{h_{i} \neq 0},{i = x},y} \\{\sigma,{h_{i} = 0}}\end{matrix} \right.} & (19)\end{matrix}$

Where h_(x), h_(y) are the control gains, n is hierarchy, g is theacceleration of gravity, Θ_(i)=Ũ₁{tilde over (H)}_(i)S_(i), i is x or y,and σ is a small constant.

It should be noted that the processing unit 150 changes the controlgains by adjusting the hierarchy n, so as to adjust the convergence timeof the gain adjustment functions (17) and (18).

Then, the processing unit 150 adjusts the target attitude parametersthrough attitude estimation functions according to the current motioninformation and the control gains (i.e. h_(x), h_(y)), where theattitude estimation functions (20), (21) are defined as follows:

$\begin{matrix}{\theta_{d} = {\frac{1}{{\overset{\sim}{U}}_{1\;}}\left( {{{- K_{x}}S_{x}} + {\overset{¨}{x}}_{d} - {k_{x}{\overset{.}{e}}_{x}} - {{h_{x}}S_{x}}} \right)}} & (20) \\{\varphi_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}\left( {{{- K_{y}}S_{y}} + {\overset{¨}{y}}_{d} - {k_{y}{\overset{.}{e}}_{y}} - {{h_{y}}S_{y}}} \right)}} & (21)\end{matrix}$

In other words, the processing unit 150 adjusts the control gainsaccording to the gain adjustment functions (17), (18), (19), so as tocontrol each set of the rotors 110 ₁-110 _(t), such that a flightattitude of the multiple rotors aircraft 100 is adjusted to the targetpitch angle θ_(d) and the target yaw angle φ_(d) , and the multiplerotors aircraft 100 may accurately fly to the target position.

{t_(v)}_(v=1) ^(∞) is conformed to a measure zero condition, such thath_(i)(t_(v))=0, v=1,2,3, . . . ∞. Moreover, through theoretical analysisof stability conditions of Lyapunov and Kurzweil theory, it is learnedthat based on the equations (15)-(20), an error of position control ofthe multiple rotors aircraft 100 may reach an asymptotically stablecondition. On the other hand, in the aforementioned descriptions, thederivation of the four-rotor aircraft is taken as an example fordescriptions, and the multiple rotors aircraft with other number of therotors can also be derived to calculate the corresponding error function(15), the gain adjustment functions (17), (18), (19) and the attitudeestimation functions (20), (21).

FIG. 5 is an example of a control flow according to an embodiment of theinvention. Referring to FIG. 5, the control method of the presentembodiment is adapted to the multiple rotors aircraft 100 of FIG. 1, inthe following description, the control method of the present embodimentis described with reference of various components and modules of themultiple rotors aircraft 100. The flow of the control method can beadjusted according to an actual requirement, which is not limited to theflow shown in FIG. 5.

The processing unit 150 obtains the current motion information (forexample, an acceleration and an angular acceleration of each axis)through the accelerometer and the gyro sensor in the inertial navigationelement 130 (step S510), and performs a transformation procedure on thecurrent motion information (step S530). The processing unit 150processes the value of the current motion information through a low-passfilter to obtain a required value, and further calculates through anintegrator to obtain current speeds along three axial directions ({dotover (x)}

{dot over (y)}

ż), the current pitch angle φ, the current yaw angle θ and the currentroll angle ψ (step S531). The processing unit 150 performs correction,transformation and estimation on the current speed ({dot over (x)}-{dotover (7)}-ż) through matrix adjustment (step S532), moving averagecalculation (step S535), a Kalman Filter (step S537) and a sliding modeobserver (step S539) to calculate the current position (x,y,z). On theother hand, the processing unit 150 performs matrix adjustment on thecurrent pitch angle φ, the current yaw angle θ and the current rollangle ψ (step S532) to transform each of the Euler angle coefficientsinto the geodesic coordinate system. Then, the processing unit 150applies the above results to the equations (15)-(20) to respectivelygenerate (for example, the error function (15), the gain adjustmentfunctions (17), (18), (19) and the attitude estimation functions (20),(21)) (step S540), such that the current position (x,y,z) is conformedto the target position (x_(d),y_(d),z_(d)) and the current attitudeparameters (i.e. the current pitch angle φ, the current yaw angle θ andthe current roll angle ψ) are conformed to the target attitudeparameters (i.e. the target pitch angle θ_(d), the target yaw angleφ_(d) and the target roll angle ψ_(d)) (steps S550, S555). Theprocessing unit 150 transforms a system input into a trust input matrixform (step S570) to drive a brushless DC motor driver (step S580). Then,the processing unit 150 transforms a trust input into the trust inputmatrix form (step S591) to control the multiple rotors aircraft 100(step S593). The aforementioned operation flow is repeatedly performedto converge the error value between the current position and the targetposition to zero.

It should be noted that according to a different design requirement, thetransformation procedure of the step S530 can be varied along withdifferent algorithms, and the embodiment of the invention is not limitedto the flow of FIG. 5.

In summary, in the multiple rotors aircraft and the control methodthereof provided by the invention, the inertial navigation element isapplied to measure the flight attitude of the multiple rotors aircraft(for example, the current pitch angle φ, the current yaw angle θ and thecurrent roll angle ψ), and the error function based on theon-Lipschitzian characteristic is adopted to derive the attitudeestimation functions and the gain adjustment functions, so as to adjustthe control gains and the flight attitude. In this way, the error valuebetween the current position and the target position may reach a fastconvergence effect according to embodiment of the invention, so as toprovide a robust adaptive control system.

It will be apparent to those skilled in the art that variousmodifications and variations can be made to the structure of theinvention without departing from the scope or spirit of the invention.In view of the foregoing, it is intended that the invention covermodifications and variations of this invention provided they fall withinthe scope of the following claims and their equivalents.

What is claimed is:
 1. A control method, adapted to a multiple rotorsaircraft, comprising: obtaining current motion information of themultiple rotors aircraft; and adjusting at least one control gainthrough a gain adjustment function according to the current motioninformation, so as to control at least one rotor of the multiple rotorsaircraft according to the at least one control gain, wherein the gainadjustment function conforms to a non-Lipschitzian characteristic. 2.The control method as claimed in claim 1, wherein the non-Lipschitziancharacteristic represents that a convergence value of a function is onlyzero, and the convergence value is not varied after being converged tozero.
 3. The control method as claimed in claim 1, wherein the currentmotion information comprises a current pitch angle, a current yaw angleand a current roll angle, and the gain adjustment function is {dot over(h)}_(x)=−h_(x) ^((1/2n+1))[Ũ₁(−θ_(d)+cos ψ sin θ cos φ+sin ψ sinφ)S_(x)] {dot over (h)}_(y)=−h_(y) ^((1/2n+1))[Ũ₁(−φ_(d)+sin ψ sin θ cosφ−cos ψ sin φ)S_(y)], where h_(x), h_(y) are the at least one${\overset{.}{h}}_{i} = \left\{ \begin{matrix}{{{- h_{i}^{({{{1/2}n} + 1})}}\Theta_{i}},{h_{i} \neq 0},{i = x},y} \\{\sigma,{h_{i} = 0}}\end{matrix} \right.$ control gain${{\overset{\sim}{U}}_{1} = {\frac{1}{\cos \; \theta \; \cos \; \varphi}\left( {g + {\overset{¨}{z}}_{d} - {k_{z}{\overset{.}{e}}_{z}} - {K_{z}S_{z}}} \right){\forall{{\theta } < {\pi/2}}}}},{{\varphi } < {\pi/2}},,$θ_(d) is a target pitch angle in at least one target attitude parameter,φ_(d) is a target yaw angle in the at least one target attitudeparameter, ψ is the current roll angle, θ is the current pitch angel, φis the current yaw angle, S_(x)=ė_(x)+k_(x)e_(x) , e_(x)=−x_(d),S_(y)=ė_(y)+k_(y)e_(y), e_(y)=x−x_(d), S_(z)=ė_(z)+k_(z)e_(z),e_(z)=z−z_(d),(x,y,z) is a current position, (x_(d),y,z_(d)) is a targetposition, k_(x), k_(y), and k_(z) are control coefficients, K_(z) is anaerodynamic damping coefficient, n is hierarchy, g is the accelerationof gravity, Θ_(i)=Ũ₁{tilde over (H)}_(i)S_(i), {tilde over(H)}_(x)=−θ_(d)+cos ψ sin θ cos φ+sin ψ sin φ, {dot over(H)}_(y)=−φ_(d)+sin ψ sin θ cos φ−cos ψ sin φ, and σ is a constant. 4.The control method as claimed in claim 3, wherein after the step ofadjusting the at least one control gain through the gain adjustmentfunction according to the current motion information, the method furthercomprises: adjusting the at least one target attitude parameter throughan attitude estimation function according to the current motioninformation and the at least one control gain, wherein the attitudeestimation function is${\theta_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}\left( {{{- K_{x}}S_{x}} + {\overset{¨}{x}}_{d} - {k_{x}{\overset{.}{e}}_{x}} - {{h_{x}}S_{x}}} \right)}},{\varphi_{d} = {\frac{1}{{\overset{\sim}{U}}_{1\;}}{\left( {{{- K_{y}}S_{y}} + {\overset{¨}{y}}_{d} - {k_{y}{\overset{.}{e}}_{y}} - {{h_{y}}S_{y}}} \right).}}}$5. The control method as claimed in claim 3, wherein before the step ofadjusting the at least one control gain through the gain adjustmentfunction according to the current motion information, the method furthercomprises: defining an error function, wherein the error function is${{\overset{.}{S}}_{\xi} = {{{\overset{\sim}{U}}_{1}{{bm}^{- 1}\begin{bmatrix}{\theta_{d} + {\overset{\sim}{H}}_{x}} \\{\varphi_{d} + {\overset{\sim}{H}}_{y}} \\{\cos \; \theta \; \cos \; \varphi}\end{bmatrix}}} + \begin{bmatrix}{{- {\overset{¨}{x}}_{d}} + {k_{x}{\overset{.}{e}}_{x}}} \\{{- {\overset{¨}{y}}_{d}} + {k_{y\;}{\overset{.}{e}}_{y}}} \\{{- {\overset{¨}{z}}_{d}} + {k_{z}{\overset{.}{e}}_{z}}}\end{bmatrix} + \begin{bmatrix}0 \\0 \\{- g}\end{bmatrix}}},$ b is a length constant, and the error functionconforms to the non-Lipschitzian characteristic.
 6. The control methodas claimed in claim 3, wherein the current motion information furthercomprises the current position, the step of obtaining the current motioninformation of the multiple rotors aircraft comprises: obtaining thecurrent position, the current pitch angle, the current yaw angle and thecurrent roll angle through at least one inertial navigation element. 7.A multiple rotors aircraft, comprising: at least one rotor, respectivelycontrolled by at least one control gain; at least one inertialnavigation element, obtaining current motion information of the multiplerotors aircraft; and a processing unit, coupled to the at least onerotor and the at least one inertial navigation element, obtaining thecurrent motion information of the multiple rotors aircraft through theat least one inertial navigation element, and adjusting the at least onecontrol gain through a gain adjustment function according to the currentmotion information, wherein the gain adjustment function conforms to anon-Lipschitzian characteristic.
 8. The multiple rotors aircraft asclaimed in claim 7, wherein the non-Lipschitzian characteristicrepresents that a convergence value of a function is only zero, and theconvergence value is not varied after being converged to zero.
 9. Themultiple rotors aircraft as claimed in claim 7, wherein the currentmotion information comprises a current pitch angle, a current yaw angleand a current roll angle, and the gain adjustment function is {dot over(h)}_(x)=−h_(x) ^((1/2n+1))[Ũ₁(−θ_(d)+cos ψ sin θ cos φ+sin ψ sinφ)S_(x)] {dot over (h)}_(y)=−h_(y) ^((1/2n+1))[Ũ₁(−φ_(d)+sin ψ sin θ cosφ−cos ψ sin φ)S], where h_(x), h_(y) are the at least one${\overset{.}{h}}_{i} = \left\{ \begin{matrix}{{{- h_{i}^{({{{1/2}n} + 1})}}\Theta_{i}},{h_{i} \neq 0},{i = x},y} \\{\sigma,{h_{i} = 0}}\end{matrix} \right.$ control gain,${{\overset{\sim}{U}}_{1} = {\frac{1}{\cos \; \theta \; \cos \; \varphi}\left( {g + {\overset{¨}{z}}_{d} - {k_{z}{\overset{.}{e}}_{z}} - {K_{z}S_{z}}} \right){\forall{{\theta } < {\pi/2}}}}},{{\varphi } < {\pi/2}},,$θ_(d) is a target pitch angle in at least one target attitude parameter,φ_(d) is a target yaw angle in the at least one target attitudeparameter, ψ is the current roll angle, θ is the current pitch angel, φis the current yaw angle, S_(x)=ė_(x)+k_(x)e_(x) , e_(x)=x−x_(d),S_(y)=ė_(y)+k_(y)e_(y), e_(y)=x−x_(d), S_(z)=ė_(z)+k_(z)e_(z),e_(z)=z−z_(d),(x,y,z) is a current position, (x_(d),y_(d),z_(d)) is atarget position, k_(x), k_(y), and k_(z) are control coefficients, K_(z)is an aerodynamic damping coefficient, n is hierarchy, g is theacceleration of gravity, Θ_(i)=Ũ₁{tilde over (H)}_(i)S_(i), {tilde over(H)}_(x)=−θ_(d)+cos ψ sin θ cos φ+sin ψ sin φ, {tilde over(H)}_(y)=−φ_(d)sin ψ sin θ cos φ−cos ψ sin φ, and σ is a micro constant.10. The multiple rotors aircraft as claimed in claim 9, wherein theprocessing unit adjusts the at least one target attitude parameterthrough an attitude estimation function according to the current motioninformation and the at least one control gain, wherein the attitudeestimation function is${\theta_{d} = {\frac{1}{{\overset{\sim}{U}}_{1}}\left( {{{- K_{x}}S_{x}} + {\overset{¨}{x}}_{d} - {k_{x}{\overset{.}{e}}_{x}} - {{h_{x}}S_{x}}} \right)}},{\varphi_{d} = {\frac{1}{{\overset{\sim}{U}}_{1\;}}{\left( {{{- K_{y}}S_{y}} + {\overset{¨}{y}}_{d} - {k_{y}{\overset{.}{e}}_{y}} - {{h_{y}}S_{y}}} \right).}}}$11. The multiple rotors aircraft as claimed in claim 9, wherein theprocessing unit defines an error function, wherein the error function is${{\overset{.}{S}}_{\xi} = {{{\overset{\sim}{U}}_{1}{{bm}^{- 1}\begin{bmatrix}{\theta_{d} + {\overset{\sim}{H}}_{x}} \\{\varphi_{d} + {\overset{\sim}{H}}_{y}} \\{\cos \; \theta \; \cos \; \varphi}\end{bmatrix}}} + \begin{bmatrix}{{- {\overset{¨}{x}}_{d}} + {k_{x}{\overset{.}{e}}_{x}}} \\{{- {\overset{¨}{y}}_{d}} + {k_{y\;}{\overset{.}{e}}_{y}}} \\{{- {\overset{¨}{z}}_{d}} + {k_{z}{\overset{.}{e}}_{z}}}\end{bmatrix} + \begin{bmatrix}0 \\0 \\{- g}\end{bmatrix}}},$ b is a length constant, and the error functionconforms to the non-Lipschitzian characteristic.